Gruber P. Convex and Discrete Geometry

344 Convex Polytopes
For x ∈ Bd ∩ H − we have xd ≥ 0 and x2 1 +···+x2 d−1 ≤ 1 − x2 d , or
Clearly,
d 2 − 1
d 2
d 2
x 2 1 +···+ d2 − 1
d2 x 2 d−1 ≤ d2 − 1
d2 (d + 1) 2 �
xd − 1
d + 1
Addition then gives
d 2 − 1
d 2
� 2
= (d + 1)2
d2 x 2 d
− d2 − 1
d2 x 2 d .
− 2(d + 1)
d 2
x 2 1 +···+d2 − 1
d2 x 2 + 1)2
d−1 +(d
d2 �
xd − 1
�2 ≤ 1+
d + 1
xd + 1
.
d2 2d + 2
d2 (x 2 d−xd) ≤ 1
on noting that 0 ≤ xd ≤ 1 implies that x 2 d − xd ≤ 0. Thus x ∈ F, concluding the
proof that B d ∩ H − ⊆ F. Finally,
V (F)
V (B d ) =
�
d2 d2 �
− 1
d−1
2 d
1
≤ e d
d + 1 2 d−1
−1 2 −
e 1
1
−
d+1 = e 2(d+1) ,
where we have used the fact that 1 + x ≤ ex for x = 1
d2 −1
,
−1 d+1 . ⊓⊔
How to Find a Feasible Solution by the Ellipsoid Algorithm?
We consider the feasible set P ={x : Ax ≤ b} of a linear optimization problem.
Assuming that there are ϱ, δ > 0 such that P ⊆ ϱB d and V (P) ≥ δ, we describe
how to find a feasible solution, i.e. a point of P.
Let E0 be the ellipsoid ϱB d with centre c0 = o. Then P ⊆ E0.Ifc0 ∈ P, then c0
is the required feasible solution. If c0 �∈ P, then c0 is not contained in at least one of
the defining halfspaces of P,sayc0�∈ {x : ai0 x ≤ βi0 }. Clearly, P ⊆ E0∩ H −
0 where
H −
0 is the halfspace {x : ai0 x ≤ ai0c0} which contains c0 on its boundary hyperplane.
By Lemma 20.1, there is an ellipsoid E1 with centre c1 such that P ⊆ E0∩ H −
and
V (E1)
1
≤ e− 2(d+1) .
V (E0)
0
⊆ E1
If c1 ∈ P, then c1 is the required feasible solution, otherwise repeat this step with
E1, c1 instead of E0, c0.
In this way we either get a feasible solution cn ∈ P in finitely many steps, or
there is a sequence of ellipsoids E0, E1, ···⊇ P with
V (En) =
V (En) V (E1)
···
V (En−1) V (E0) V (E0)
n
−
≤ e 2(d+1) V (Eo) for n = 1, 2,...
Thus V (En)